What is the Pythagorean theorem?

The **Pythagorean theorem** is a fundamental principle in geometry that relates the lengths of the sides of a right-angled triangle. It states that: > In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, if a triangle has sides of length \(a\) and \(b\) forming the right angle, and hypotenuse length \(c\), then: \[ c^2 = a^2 + b^2 \] ### Explanation This theorem allows you to calculate the length of one side of a right triangle if you know the other two. It is widely used in various fields such as architecture, engineering, physics, and even computer graphics. ### Proof (One of Many) One classic geometric proof involves rearranging areas: 1. Construct a square with side length \(a + b\). 2. Inside this square, arrange four copies of the right triangle so that their hypotenuses form a smaller, tilted square in the center. 3. The area of the large square is \((a + b)^2\). 4. The four triangles each have area \(\frac{1}{2}ab\), so together they cover \(2ab\). 5. The remaining inner square, whose side length is \(c\), has area \(c^2\). 6. Therefore, the total area can also be expressed as the sum of the areas of the four triangles plus the inner square: \[ (a + b)^2 = 4 \times \frac{1}{2}ab + c^2 = 2ab + c^2 \] 7. Expanding the left side: \[ a^2 + 2ab + b^2 = 2ab + c^2 \] 8. Subtracting \(2ab\) from both sides gives: \[ a^2 + b^2 = c^2 \] This geometric rearrangement visually demonstrates the truth of the Pythagorean theorem. --- If you want, I can also provide algebraic proofs or explore applications of the theorem!

The **Pythagorean theorem** is a fundamental principle in geometry that relates the lengths of the sides of a right-angled triangle. It states that: > **In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.** Mathematically, if a triangle has sides \(a\) and \(b\) forming the right angle, and \(c\) is the hypotenuse, then: \[ c^2 = a^2 + b^2 \] ### Explanation This theorem allows you to calculate the length of one side of a right triangle if you know the lengths of the other two. It’s widely used in various fields such as architecture, engineering, physics, and even computer graphics. ### Proof (Using a Geometric Approach) One classic proof uses the concept of rearranging areas: 1. Construct a large square with side length \(a + b\). 2. Inside this large square, place four copies of the right triangle with sides \(a\), \(b\), and \(c\), arranged so that their hypotenuses form a smaller, tilted square in the center. 3. The area of the large square is \((a + b)^2\). 4. The four triangles together have a total area of \(4 \times \frac{1}{2}ab = 2ab\). 5. The remaining area inside the large square (the smaller square formed by the hypotenuses) has area \(c^2\). Since the large square’s area is made up of the four triangles plus the smaller square, you can write: \[ (a + b)^2 = 4 \times \frac{1}{2}ab + c^2 \] Simplify: \[ a^2 + 2ab + b^2 = 2ab + c^2 \] Subtract \(2ab\) from both sides: \[ a^2 + b^2 = c^2 \] This completes the proof. There are numerous other proofs of the Pythagorean theorem, including algebraic and trigonometric ones, but this geometric proof is one of the most intuitive and widely taught.